An important issue in the ED reconstruction of quantum is whether ED is, in fact, actually equivalent to quantum theory. While this issue was first raised by T.C. Wallstrom [1]} in the context of Nelson's stochastic mechanics, it poses a significant challenge to any reconstruction of quantum theory whose principal variables are expressed in terms of probabilities and phases, rather than wave functions; these include, of course, Nelson's stochastic mechanics [2]}, but also, for instance, the hydrodynamical formulations of Takabayasi [3]} and Madelung [4]}. In short, Wallstrom's objection is as follows. Conventional quantum theory requires the wave function \(\Psi \)  to be single valued, all while allowing its phase \(\Phi \)  to be a multi-valued function. The difficulty faced by ED, and those theories formally similar to ED, is that the phase function \(\Phi \)  is either too restricted — it is singled valued — or when \(\Phi \)  is allowed to be multi-valued, there is no plausible rationale for requiring \(\Psi \)  to then be single valued. The former case leads to a rather stunted quantum theory where crucial states of non-zero angular momentum, for example, are excluded, whereas the latter case is too flexible; the total angular momentum is not quantized, which disagrees with the empirical evidence. In either case, the theories do not reproduce the physical Hilbert space of solutions from the standard theory.
